score:3
Chesterfield appears not to have much in the way of car rental agencies. That aside, here's a comparison of various towns dotted outside the Peak District:
From London (train) To Bakewell (car) Walk to Europcar Walk to Hertz
Birmingham 1 hr 25 1 hr 35 10 mins 15 mins
Manchester 2 hr 15 1 hr 10 15 mins 0 mins
Nottingham 1 hr 50 1 hr 10 mins 20 mins
Sheffield 2 hr 15 40 mins 20 mins 10 mins
Derby 1 hr 30 50 mins 25 mins 35 mins (10 mins bus)
Stafford 1 hr 15 80 mins
Stoke-on-Trent 1 hr 30 60 mins
Stockport 1 hr 55 50 mins
Since it looks like a car rental agency can always be found within 10 minutes of walking or public transport from the train station, Derby looks to be the winner as it's both easily reached by train and less than an hour's drive from the centre of the Peak District.
I'm open to running numbers on other locations if suggested. These are just culled from scouting around the surrounds of the Peak District on Google Maps.